Algebraic Topology II: Topological KTheory (Spring 2015)
This course is taught jointly by
Javier J. Gutiérrez,
Ieke Moerdijk, and
Matan Prasma.
The teaching assistant for this course is
Joost Nuiten.

Time and place:
Thursdays 10:3013:30, Radboud Universiteit Nijmegen, room HG03.085 (Huygensgebouw).
Course description >
Topological
Ktheory, the first generalized cohomology theory to be studied thoroughly,
was introduced in a 1961 paper by Atiyah and Hirzebruch, where they adapted the work
of Grothendieck on algebraic varieties to a topological setting. Since that time, topological
Ktheory has become a powerful and indispensable tool in topology, differential geometry,
and index theory.
We will begin by studying the theory of vector bundles and related algebraic notions, followed by
the definition of
Ktheory and proofs of some important theorems in the subject, such as the Bott periodicity theorem.
Some of the topics that might be covered in the course include:

Basics about vector bundles. Definition of K(X), K^{n}(X).

Homotopy invariance and long exact sequence

EilenbergSteenrod axioms

Operations with vector bundles

Classifying space of a vector bundle

Cohomology of the infinite Grassmanians

Adams operations and lambdarings
Lecture notes >

Lecture 6: Ktheory as a cohomology theory. (12/03)

Lecture 7: Ktheory groups of the spheres. (19/03)

Paper by B. Harris,
Bott periodicity via simplicial spaces, J. Algebra, 62 (1980), 450–454.

Paper by I. Moerdijk,
Bisimplicial sets and the groupcompletion theorem, Algebraic Ktheory: connections with geometry and topology (Lake Louise, AB, 1987), 225–240, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 279, 1989.
Exercise sheets >
References >
 M. F. Atiyah, Ktheory. Lecture notes by D. W. Anderson W. A. Benjamin, Inc., New YorkAmsterdam 1967
 A. Hatcher, Vector bundles and Ktheory, Book project available at the author's webpage.
 D. Husemoller, Fibre bundles. Graduate Texts in Mathematics, 20. SpringerVerlag, New York, 1994.
 M. Karoubi, Ktheory. An introduction. Grundlehren der Mathematischen Wissenschaften, Band 226. SpringerVerlag, BerlinNew York, 1978.
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